A Family of Self-Dual Quasicrystals with Critical Phases

Abstract: We propose a general framework for constructing self-dual one-dimensional quasiperiodic lattice models with arbitrary-range hoppings and multifractal behaviors. Our framework generates a broad spectrum of quasicrystals, ranging from the off-diagonal Aubry-Andr\'e-Harper models on one end, to those with long-range power-law hoppings on another. Focusing on models with off-diagonal quasiperiodic hoppings with power-law decay, we exploit the fact that, when the self-dual condition is satisfied, the system must be in the critical state with multifractal properties. This enables the engineering of models with competing extended, critical, and localized phases, with rich mobility edges in between. As an outstanding example, we show that a limiting case of our family of self-dual quasicrystals can be implemented using Rydberg-atom arrays. Our work offers a systematic route toward self-duality and critical phases, and would facilitate their experimental simulation.